By: Melissa Wilson
In this assignment we will investigate values for a in:
We will look at various values for a: -5, -3, -1, 1, 3, 5
First let's look at the difference between positive and negative coefficients. All of the equations with positive coefficients open upward (concave up) and the negative coefficients all point downward (concave down). Since the x-term is squared, for any value of x, that term is positive. Therefore, the coefficient solely determines the sign for the y value. Hence, the positive a values will give all positive values for y and and negative a values will make all the y values negative.
Now, look at the difference in the coefficients based on their scale. As the absolute value of the a coefficient increases the width of the parabola gets smaller. This is because as the a increases, the corresponding y-value increases. For instance, let's compare the value at x=2 for two separate equations. First,
Now, for another,
We can easily see that a coefficient of 3 will increase the y-value at a given x by 3 times. Hence, the y-values for the parabola will increase more rapidly as the absolute value of a increases.
Below you will see an animation of
with n varying from -10 to 10.
